Институт вычислительной математики и математической геофизики СОРАН

Abstracts

Statistical simulation and Monte Carlo method

KEYWORDS: financial modelling, occupation time, solvable diffusion processes, exact simulation, bridge sampling, first hitting time, Bessel processes, CIR process, CEV model, confluent hypergeometric diffusions

This talk is twofold. First, we consider the simulation of the squared Bessel process and some other continuous-time one-dimensional stochastic processes such as the CIR process, constant elasticity of variance (CEV) diffusion model, and hypergeometric diffusions, which can all be obtained from a Bessel process by using a change of variable, time and scale transformations, and/or a change of measure. All these processes are broadly used in mathematical finance for modelling asset values and interest rates. The object of our interest is the precise path sampling of the abovementioned stochastic processes from their exact multivariate distributions. We show how the transition and bridge probability distributions of a Bessel process with or without absorption at the origin reduce to so-called randomized gamma distributions.

In the second part, we present new simulation algorithms and analytical methods for pricing occupation-time derivatives. As an underlying asset price process we consider a jump-diffusion process with finite jump intensity or a solvable nonlinear diffusion model. For Brownian motion with a Poisson component, we obtain an exact simulation algorithm. For nonlinear diffusion models, the occupation time is simulated approximately by using the linear interpolation, first hitting time method, or Brownian bridge approximation. In the end, we present analytical evaluation methods which are based on numerical Laplace inversion transform.

Note. Abstracts are published in author's edition

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